Integrand size = 22, antiderivative size = 51 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x} \, dx=3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+3 b n \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2441, 2352} \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x} \, dx=3 \log \left (-\frac {e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+3 b n \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{x} e}{d}+1\right ) \]
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Rule 2352
Rule 2441
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )-(3 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+3 b n \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x} \, dx=3 b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+a \log (x)+3 b n \operatorname {PolyLog}\left (2,\frac {d+e \sqrt [3]{x}}{d}\right ) \]
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\[\int \frac {a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )}{x}d x\]
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\[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a}{x} \,d x } \]
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\[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x} \, dx=\int \frac {a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{x}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (44) = 88\).
Time = 0.34 (sec) , antiderivative size = 166, normalized size of antiderivative = 3.25 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x} \, dx=-3 \, {\left (\log \left (\frac {e x^{\frac {1}{3}}}{d} + 1\right ) \log \left (x^{\frac {1}{3}}\right ) + {\rm Li}_2\left (-\frac {e x^{\frac {1}{3}}}{d}\right )\right )} b n + \frac {4 \, b d^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n}\right ) \log \left (x\right ) + 4 \, {\left (b d^{2} \log \left (c\right ) + a d^{2}\right )} \log \left (x\right ) + \frac {2 \, b e^{2} n x \log \left (x\right ) - 3 \, b e^{2} n x}{x^{\frac {1}{3}}} - \frac {4 \, {\left (b d e n x \log \left (x\right ) - 3 \, b d e n x\right )}}{x^{\frac {2}{3}}}}{4 \, d^{2}} + \frac {3 \, {\left (b e^{2} n x^{\frac {2}{3}} - 4 \, b d e n x^{\frac {1}{3}} - 2 \, {\left (b e^{2} n x^{\frac {2}{3}} - 2 \, b d e n x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3}}\right )\right )}}{4 \, d^{2}} \]
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\[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x} \, dx=\int { \frac {b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a}{x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{x} \,d x \]
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